A309319 E.g.f.: 1/Product_{k>0} (1 - x^(2*k)/(2*k)) (even powers only).

1, 1, 12, 300, 15960, 1232280, 157006080, 25418352960, 5859886032000, 1655203620470400, 604893737678630400, 261278195494386470400, 140231830875916632652800, 86107922772424330377600000, 63316800257542340301112320000, 52666943508290765740968161280000

Offset: 0

Author

Vaclav Kotesovec, Jul 23 2019

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..225
• D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 6).

Crossrefs

Cf. A007838, A007841, A087639, A088994, A294506.

Programs

Mathematica

nmax = 20; Table[(CoefficientList[Series[1/Product[(1 - x^(2*k)/(2*k)), {k, 1, 2*nmax}], {x, 0, 2*nmax}], x]*Range[0, 2*nmax]!)[[2 n + 1]], {n, 0, nmax}]

Formula

a(n) ~ exp(-gamma/2) * (2*n)! / sqrt(n) [Lehmer, 1972], where gamma is the Euler-Mascheroni constant A001620.